Seder Olam according to this translation states that Yoseph died at 110 in the year 2309. The translator also states that Levi (the last of the brothers to die) lived from 2194 - 2331 (year 95 in Mitzrayim). He dates the bondage from then. Note that given the 116 years mentioned in the quote below, the bondage would start from 2332. This may be a rounding error or that the bondage started the year after Levi's death.
However, based on the fact that Yoseph had to return from the burial of his father (and the children were kept as hostages) they were not allowed to leave Mitzrayim from that time or earlier. They may not have been allowed to leave from the time Yaakov came in 2238, since the famine ended then and Par'o would not have wanted to allow the bracha to depart (and Yoseph was still in power until he died). This means while the bondage lasted 116 years, they were restricted from leaving from the time that Yoseph died.
Note that the table derived from this seems to be mistaken as to the birth of Moshe and the date of the Exodus as the translator counts 216 years in Mitzrayim rather than 210 and dates the Exodus as 2454 rather than 2448 and the birth of Moses as 2374 rather than 2368.
And during all the years that Levi was alive, Israel was not enslaved
in Egypt, as it is said: (Exodus 1:6) And Joseph died and all his
brothers etc. and (Exodus 1:8) A new king rose etc. and, from the
moment Levi died, the Egyptians started to enslave them. From this
they said [when] one of the brothers dies, all the brothers will take
care [of his tribe], one of the group dies, all the group will take
care. (Exodus 1:7-8) And the children of Israel were fruitful and
increased etc. And a new king rose etc. It is found that from the
moment Levi died until Israel came out of Egypt was 116 years. And
the bondage was no more, and no less than 86 years as being the years
of Miriam [in Egypt]. And why was she called Miriam? Because of the
By the time that the sh'vatim died, they had increased enough that it would have caused a major hit to the economy had they left. Also, there were too many to actually all leave as one unit and be able to survive the trek as well as to settle back in Canaan without being seen as an invasion (like the 10,000 of Ephraim that tried to leave and wound up as the valley of the dried bones). Note that Par'0 kept the wives and children as hostages when they went to bury Ya'akov.
Rav Hirsch says (Shmos 1:8) that a new dynasty overthrew the government after the death of Yosef (and his brothers) and wanted to use propaganda to make sure that the Egyptians would not rebel. They were afraid that the Jews (who were an independent people) would trigger a revolt using the gratitude that the native Egyptians (who still remembered how Yosef had saved them) still had for the Jews.
The Egyptians were already subjugated, and it was to his own people
that the King spoke: "We need not fear the Egyptians, they are already
under our power, but in this outlying province, a race is growing that
may become too strong for us, whom we may not be able to conquer so
Egypt was divided into castes, consisted of different unrelated עמים
so the King could well say: See, these Jews who still hang together as
one nefesh, they will become too powerful for us, there is no single
caste which is as numerous, which has such an amount of potential
power as they have.
Rav Hirsch goes into more detail how Par'o wanted to create a group that everyone else could look down on and forget that he was keeping them enslaved and oppressed.